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arxiv: 2604.18389 · v1 · submitted 2026-04-20 · 💻 cs.CL

Recognition: unknown

Understanding the Prompt Sensitivity

Yang Liu , Chenhui Chu
Authors on Pith no claims yet

Pith reviewed 2026-05-10 04:37 UTC · model grok-4.3

classification 💻 cs.CL
keywords prompt sensitivitylarge language modelsTaylor expansionCauchy-Schwarz inequalitylog probabilitymeaning-preserving promptslogit variance
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The pith

Large language models disperse meaning-preserving prompts rather than clustering them, which produces an excessively high upper bound on next-token log-probability differences.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper treats large language models as multivariate functions and applies a first-order Taylor expansion to relate small changes in prompts to changes in next-token log probabilities. It then applies the Cauchy-Schwarz inequality to obtain an upper bound on the difference in those probabilities for any two prompts that preserve meaning. The central finding is that the models disperse similar inputs across their input space instead of clustering them internally, as smaller networks do; this dispersion inflates the bound and prevents it from being driven close to zero. The work further shows that certain categories of meaning-preserving variants carry higher risk and that the bound itself tracks an existing sensitivity score while prompt templates dominate logit variance over the questions. A sympathetic reader would care because the result supplies a concrete mechanism for why LLMs change outputs under rewordings that leave meaning unchanged.

Core claim

By modeling the LLM as a function and expanding the next-token log-probability around a base prompt to first order, the difference for a meaning-preserving variant is bounded above by the product of the gradient norm and the prompt difference norm via Cauchy-Schwarz. The analysis establishes that LLMs disperse rather than cluster similar inputs, yielding an upper bound that remains large and difficult to reduce to zero. Prompt templates are shown to exert stronger influence on logits than the questions themselves, and the bound correlates with PromptSensiScore while identifying higher-risk variant types.

What carries the argument

The first-order Taylor expansion of the next-token log-probability function together with the Cauchy-Schwarz inequality, which together bound probability differences in terms of input dispersion.

If this is right

  • Certain categories of meaning-preserving prompt variants carry higher risk of introducing sensitivity.
  • The derived upper bound correlates strongly with the existing PromptSensiScore metric.
  • Prompt templates exert greater influence on logits than the questions themselves.
  • The dispersing behavior prevents the log-probability difference from being reduced effectively to zero.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Techniques that reduce dispersion of equivalent inputs in gradient space could lower sensitivity without changing model weights.
  • The same expansion and bound could be applied to diagnose sensitivity in other autoregressive sequence models.
  • Measuring dispersion directly might provide a pre-deployment check for prompt stability on new models or domains.

Load-bearing premise

The first-order Taylor expansion around a prompt sufficiently approximates the relationship between meaning-preserving prompt variants and changes in next-token log probabilities for the derived bound to be informative.

What would settle it

A direct measurement of internal representations or gradients showing that meaning-preserving prompts cluster tightly rather than disperse, or a set of prompt pairs where the observed log-probability difference consistently exceeds the derived upper bound.

Figures

Figures reproduced from arXiv: 2604.18389 by Chenhui Chu, Yang Liu.

Figure 1
Figure 1. Figure 1: Intra-class mean distances for CIFAR-10 across different training epochs. where C is the class set and |C| denotes the total number of classes. Enhancing intra-class compact￾ness, i.e., decreasing Dintra, can improve the neu￾ral network’s classification performance (Liu et al., 2016; Yan et al., 2020). Intra-class mean distance of ResNet on CIFAR￾10. To illustrate the internal behavior of neural networks w… view at source ↗
Figure 2
Figure 2. Figure 2: Key results of RQ1: (a) indicates the trend [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Key results of RQ2: (a), (b), and (c) indicate the trend of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The relationship between the average upper bound across layers of LLMs and PSS. The lines represent [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of contribution rates of prompt [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The feature maps’ shape of the neural network. The batch size dimension is omitted and “ [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 13
Figure 13. Figure 13: Activation steering on Qwen1.5-4B (ARC Challenge). Bars show the mean |∆ log π(yt | h)| be￾fore (red) and after (green) steering at three layer depths l ∈ {L/4, L/2, 3L/4}. Steering consistently reduces prompt sensitivity, and the reduction grows as the intervention layer goes deeper. For example, on Qwen1.5-4B with the ARC Challenge, the baseline |∆ log π(yt | h)| is 2.49; steering at L/4, L/2, and 3L/4 … view at source ↗
Figure 7
Figure 7. Figure 7: The trends of ∥∇h log π(yt|h0)∥ and ∥∆h∥ across layers for all models on the four datasets [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The trends of upper bounds across layers for all models on the four datasets. [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The comparison results between “First” and “Latter” across all models and datasets. [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The comparison results between “Fewer” and “More” across all models and datasets. [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The comparison results between “Modify” and “Misalign” across all models and datasets. [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The comparison results between “Paraphrase,” “Typo,” and “Orthographic” across all models and [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: The comparison of the upper bound ∥∇h log π(yt|h0)∥ · ∥∆h∥ of different yt [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Activation steering across all 11 models on the four MCQ datasets. Each cell shows [PITH_FULL_IMAGE:figures/full_fig_p026_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The comparison of the absolute difference of the log probabilities [PITH_FULL_IMAGE:figures/full_fig_p027_16.png] view at source ↗
read the original abstract

Prompt sensitivity, which refers to how strongly the output of a large language model (LLM) depends on the exact wording of its input prompt, raises concerns among users about the LLM's stability and reliability. In this work, we consider LLMs as multivariate functions and perform a first-order Taylor expansion, thereby analyzing the relationship between meaning-preserving prompts, their gradients, and the log probabilities of the model's next token. We derive an upper bound on the difference between log probabilities using the Cauchy-Schwarz inequality. We show that LLMs do not internally cluster similar inputs like smaller neural networks do, but instead disperse them. This dispersing behavior leads to an excessively high upper bound on the difference of log probabilities between two meaning-preserving prompts, making it difficult to effectively reduce to 0. In our analysis, we also show which types of meaning-preserving prompt variants are more likely to introduce prompt sensitivity risks in LLMs. In addition, we demonstrate that the upper bound is strongly correlated with an existing prompt sensitivity metric, PromptSensiScore. Moreover, by analyzing the logit variance, we find that prompt templates typically exert a greater influence on logits than the questions themselves. Overall, our results provide a general interpretation for why current LLMs can be highly sensitive to prompts with the same meaning, offering crucial evidence for understanding the prompt sensitivity of LLMs. Code for experiments is available at https://github.com/ku-nlp/Understanding_the_Prompt_Sensitivity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper models LLMs as multivariate functions and applies a first-order Taylor expansion around a prompt to relate meaning-preserving variants to changes in next-token log probabilities. It derives an upper bound on |Δlog p| via the Cauchy-Schwarz inequality, claims that LLMs disperse similar inputs (unlike smaller networks) producing an excessively high bound that cannot be reduced to zero, shows this bound correlates with PromptSensiScore, and reports that prompt templates exert greater influence on logits than the questions themselves.

Significance. If the linear approximation is valid and the dispersion claim is substantiated, the work supplies a general theoretical account of prompt sensitivity grounded in gradient behavior and logit variance, with the reported correlation to an existing metric and public code as reproducibility strengths. This could guide prompt design and stability analysis, though the significance depends on whether the bound is shown to be tight rather than merely mathematical.

major comments (2)
  1. [Derivation of upper bound] The derivation section (first-order Taylor expansion and Cauchy-Schwarz application): the claim that dispersing gradients produce an 'excessively high' upper bound preventing reduction to zero requires that the linear term dominate the remainder for discrete prompt edits. No quantitative bound on the Taylor remainder or empirical check of approximation error for typical meaning-preserving variants is provided, leaving open whether the bound characterizes actual Δlog p or is merely loose.
  2. [Empirical validation] The correlation analysis with PromptSensiScore: the paper presents the upper bound as strongly correlated with the metric but supplies no details on sample size, statistical significance testing, or controls, which weakens its role as supporting evidence for the dispersion explanation.
minor comments (2)
  1. [Analysis of input dispersion] Clarify the precise definition and measurement of 'dispersing behavior' versus clustering in smaller networks, including any distance metric or visualization used.
  2. [Logit variance analysis] The logit variance comparison between templates and questions would benefit from explicit statistical tests or effect sizes to support the claim that templates exert greater influence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below and will revise the paper to incorporate additional empirical checks and statistical details where appropriate, while preserving the core theoretical contribution on gradient dispersion in LLMs.

read point-by-point responses

  1. Referee: [Derivation of upper bound] The derivation section (first-order Taylor expansion and Cauchy-Schwarz application): the claim that dispersing gradients produce an 'excessively high' upper bound preventing reduction to zero requires that the linear term dominate the remainder for discrete prompt edits. No quantitative bound on the Taylor remainder or empirical check of approximation error for typical meaning-preserving variants is provided, leaving open whether the bound characterizes actual Δlog p or is merely loose.

    Authors: We acknowledge that the first-order Taylor expansion is an approximation and that the remainder term may not be negligible for discrete prompt edits. Our derivation intentionally uses the linear term to obtain a closed-form upper bound via Cauchy-Schwarz that highlights the effect of gradient dispersion; the bound is not presented as a tight characterization of every observed Δlog p but as a theoretical explanation for why sensitivity persists even when prompts are meaning-preserving. The dispersion claim itself is supported by direct measurements of gradient and embedding distances in the paper. In the revision we will add a new subsection with quantitative checks: we will compute the actual Δlog p versus the linear approximation error for representative meaning-preserving variants (e.g., paraphrases and template swaps) across several models and report the relative magnitude of the remainder, thereby clarifying the regime in which the bound remains informative. revision: partial

  2. Referee: [Empirical validation] The correlation analysis with PromptSensiScore: the paper presents the upper bound as strongly correlated with the metric but supplies no details on sample size, statistical significance testing, or controls, which weakens its role as supporting evidence for the dispersion explanation.

    Authors: We agree that additional statistical rigor is needed. The reported correlation was obtained over a collection of LLMs and prompt pairs, yet the manuscript omitted the precise sample size, correlation coefficient with p-value, and any baseline controls. In the revised version we will explicitly state the number of prompt pairs and models used, report Pearson/Spearman coefficients together with p-values, and include controls such as (i) correlations with randomly perturbed (non-meaning-preserving) prompts and (ii) comparisons against logit variance alone. These additions will strengthen the evidential link between the derived bound and observed prompt sensitivity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard inequalities on empirical dispersion observations

full rationale

The paper treats the LLM as a multivariate function f, applies a first-order Taylor expansion around a prompt, and invokes the Cauchy-Schwarz inequality to obtain an upper bound on |Δlog p| between meaning-preserving variants. This bound is a direct mathematical consequence of the expansion and inequality, independent of any fitted quantities or self-citations. Dispersion of similar inputs is reported as an empirical observation (via internal activations or gradients), which is then interpreted as producing a loose bound; the correlation with PromptSensiScore is offered only as supporting evidence. No equation reduces to a definitional identity, no parameter is fitted on a subset and renamed a prediction, and no uniqueness theorem or ansatz is smuggled via self-citation. The central claim remains an interpretive link between observed dispersion and bound looseness rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of a first-order Taylor expansion to the LLM output function and on the validity of treating prompt variants as small perturbations in input space; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption First-order Taylor expansion sufficiently approximates the change in next-token log probability for meaning-preserving prompt perturbations
    Invoked to relate gradients to output differences before applying Cauchy-Schwarz.

pith-pipeline@v0.9.0 · 5545 in / 1274 out tokens · 35956 ms · 2026-05-10T04:37:42.901350+00:00 · methodology

discussion (0)

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